Distance-Two Colouring of Graphs
JAN VAN DEN HEUVEL
joint work with
FREDERIC HAVET, COLIN MCDIARMID & BRUCE REED
and with OMID AMINI & LOUIS ESPERET
Department of MathematicsLondon School of Economics and Political Science
The start
on way to look at vertex-colouring :
vertices at distance one must receive different colours
suppose we require vertices at larger distances
to receive different colours as well
for today we only look at distance two
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
The square of a graph
distance-two colouring can be modelled using the
square G2 of a graph :
same vertex set as G
edges between vertices with distance at most 2 in G
( = are adjacent or have a common neighbour )
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Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Colouring the square of a graph
Easy facts
χ(G2) ≥ ∆(G) + 1
and
∆(G2) ≤ ∆(G)2 , so χ(G2) ≤ ∆(G)2 + 1
Question
is the upper bound relevant ?
there are at most 4 graphs with χ(G2) = ∆(G)2 + 1
and infinitely many graphs with
χ(G2) = ∆(G)2 − ∆(G) + 1
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
The square of planar graphs
Conjecture ( Wegner, 1977 )
G planar
=⇒ χ(G2) ≤
7, if ∆ = 3
∆ + 5, if 4 ≤ ∆ ≤ 7⌊3
2 ∆⌋
+ 1, if ∆ ≥ 8
bounds would be best possible
r r r p p p p r
rrrppppr
rrrp
pppr
t
t
tk − 1
k
k
case ∆ = 2 k ≥ 8 :
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
What is known for large ∆∆∆
G planar =⇒
χ(G2) ≤ 8 ∆ − 22 ( Jonas, PhD, 1993 )
χ(G2) ≤ 3 ∆ + 5 ( Wong, MSc, 1996 )
χ(G2) ≤ 2 ∆ + 25 ( vdH & McGuinness, 2003 )
χ(G2) ≤⌈9
5 ∆⌉
+ 1 ( for ∆ ≥ 47 )
( Borodin, Broersma, Glebov & vdH, 2001 )
χ(G2) ≤⌈5
3 ∆⌉
+ 24 ( for ∆ ≥ 241 )
( Molloy & Salavatipour, 2005 )
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
First new results
Theorem
G planar =⇒ χ(G2) ≤(3
2 + o(1))
∆ (∆ → ∞)
we actually prove the list-colouring version for much larger
classes of graphs :
Theorem
graph G K 3,k -minor free for some fixed k
=⇒ ch(G2) ≤(3
2 + o(1))
∆
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Even more general results ?
Property
graph G H -minor free for some fixed graph H
=⇒ ch(G2) ≤ CH ∆ for some constant CH
Question
given H, what is the best CH for large ∆?
e.g.
for H = K 5 we have 2 ≤ CK 5 ≤ 9
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
The clique number
Corollary
graph G K 3,k -minor free for some fixed k
=⇒ ω(G2) ≤(3
2 + o(1))
∆
this can be partially improved to
Theorem
G planar =⇒ ω(G2) ≤ 32 ∆ + O(1)
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
A related (?) problem
plane graph : planar graph with a given embedding
cyclic colouring of a plane graph :
vertex-colouring so that
vertices incident to the same face get a different colour
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
A related (?) problem
plane graph : planar graph with a given embedding
cyclic colouring of a plane graph :
vertex-colouring so that
vertices incident to the same face get a different colour
cyclic chromatic number χ∗(G) :
minimum number of colours needed for a cyclic colouring
∆∗(G) : size of largest face of G
Easy
G plane graph =⇒ χ∗(G) ≥ ∆∗(G)
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Conjecture for the related(?) problem
Conjecture ( Borodin, 1984 )
G plane graph =⇒ χ∗(G) ≤ 32 ∆∗(G)
bound would be best possible
r r r p p p p r
rrrppppr
rrrp
pppr
k vertices
k vertices
k vertices
case ∆∗ = 2 k :
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Bounds on the cyclic chromatic number
G plane graph =⇒
χ∗(G) ≤ 2 ∆∗(G) ( Ore & Plummer, 1969 )
χ∗(G) ≤ 95 ∆∗(G) ( Borodin, Sanders & Zhao, 1999 )
χ∗(G) ≤ 53 ∆∗(G) ( Sanders & Zhao, 2001 )
Theorem
G plane graph
=⇒ χ∗(G) ≤(3
2 + o(1))
∆∗ (∆∗ → ∞)
( in fact, we again prove the list-colouring version )
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
From cyclic colouring to “distance-two” colouring
given a plane graph G , form new graph GF by
adding a vertex in each face
adding edges from new vertex to all vertices of the face
G
-
GF
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
From cyclic colouring to “square” colouring
G
-
GF
colouring the square of GF gives cyclic colouring of G
but also colours the faces ( not asked for, but o.k. )
and degrees of old vertices may be more than ∆∗(G)
( serious problem )
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Both in one go
idea : do not treat all vertices equal :
some vertices need to be coloured
some vertices determine distance two
vertices can be of both types
as are all vertices when colouring the square
or the types can be disjoint
as for “faces” and “vertices” for cyclic colouring
or anything in between
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Formalising our clever little idea
given : graph G , subsets A, B ⊆ V(G)
(A, B)-colouring of G : colouring of vertices in B so that
adjacent vertices get different colours
vertices with a common neighbour in A
get different colours
list (A, B)-colouring of G :
similar, but each vertex in B has its own list
χ(G; A, B) / ch(G; A, B) :
minimum number of colours needed /
minimum size of each list neededDistance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
The first baby steps
χ(G) = χ(G; ∅, V)
χ(G2) = χ(G; V , V)
χ∗(G) = χ(GF ; “faces”, “vertices”)
relevant “degree” : dB(v) = number of neighbours in B
“maximum degree” :
∆(G; A, B) = maximum of dB(v) over all v ∈ A
Easy
∆(G; A, B) ≤ χ(G; A, B)
≤ ∆(G) + ∆(G; A, B) · (∆(G; A, B) − 1) + 1Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
The obvious(?) conjecture & results
Conjecture
G planar, A, B ⊆ V(G) , ∆(G; A, B) large enough
=⇒ χ(G; A, B) ≤ 32 ∆(G; A, B) + 1
Theorem
G planar, A, B ⊆ V(G)
=⇒ ch(G; A, B) ≤(3
2 + o(1))
∆(G; A, B)
Corollary( asymptotic list version of Wegner’s and Borodin’s Conjecture )
G planar =⇒ ch(G2) ≤(3
2 + o(1))
∆(G)
G plane graph =⇒ ch∗(G) ≤(3
2 + o(1))
∆∗(G)
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Sketch of the proof of square of planar graph
uses induction on the number of vertices
2-neighbour : vertex at distance one or two
d 2(v ) : number of 2-neighbours of v
= number of neighbours of v in G2
we would like to remove a vertex v with d2(v) ≤ 32 ∆
but that can change distances in G − v
contraction to a neighbour u will solve the distance problem
but may increase maximum degree if d(u) + d(v) > ∆
easy induction possible if there is an edge uv
with d(u) + d(v) ≤ ∆ and d2(v) ≤ 32 ∆
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
When easy induction is not possible
S , small vertices : degree at most some constant C
B , big vertices : degree more than C
H , huge vertices : degree at least 12 ∆
small vertices have a least two big neighbours
( otherwise for those v : d2(v) ≤ 32 ∆ )
a planar graph has fewer than 3 |V | edges
and fewer than 2 |V | edges if it is bipartite
so :
all but O(|V |/C) vertices are small
fewer than 2 |B| vertices in V \ B
have more than two neighbours in BDistance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
When easy induction is not possible
S , small vertices : degree at most some constant C
B , big vertices : degree more than C
H , huge vertices : degree at least 12 ∆
so :
“most” vertices are small
and these have exactly two big neighbours
( in fact two huge neighbours )
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
The structure so far
there is a subgraph F of G looking like :
green vertices Xhave degree at least 1
2 ∆
black vertices Yhave degree at most C F
all other neighbours of Y -vertices are also small
we can guarantee additionally :
only “few” edges from X to rest of G
F satisfies “some edge density condition”Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
The other induction step
remove the vertices from Y ( using contraction )
colour the smaller graph ( which is possible by induction )
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
The other induction step
remove the vertices from Y ( using contraction )
colour the smaller graph ( which is possible by induction )
what to do with the uncoloured Y -vertices ?
yx1 x2
y has
a “lot” of 2-neighbours in Y via x1 , x2
at most C2 other 2-neighbours in Y
at most (dG(x1) − dF(x1)) + (dG(x2) − dF(x2))
2-neighbours outside Y via x1 , x2
at most C2 other 2-neighbours outside YDistance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Transferring to edge-colouring
so a vertex y from Y has at least
(32 + ε) ∆ − (dG(x1) − dF(x1)) − (dG(x2) − dF(x2)) − C2
colours still available
colouring Y is “almost” like list-colouring edges of the
multigraph F e :
-
F F e
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Edge-colouring multigraphs
χ′(G) : chromatic index of multigraph G
ch’ (G) : list chromatic index of multigraph G
χ′F (G) : fractional chromatic index of multigraph G
Theorem ( Kahn, 1996, 2000 )
G multigraph, with ∆ large enough
=⇒ ch′(G) ≈ χ′(G) ≈ χ′F(G)
in fact, Kahn’s proofs provide something much more general
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Kahn’s result
Theorem ( Kahn, 2000 )
for 0 < δ < 1, α > 0 there exists ∆δ,α so that if ∆ ≥ ∆δ,α :
G a multigraph with maximum degree ∆
each edge e has a list L(e) of colours so that
for all edges e : |L(e)| ≥ α ∆
for all vertices v :∑
e∋v|L(e)|−1 ≤ 1 · (1 − δ)
for all K ⊆ G with |V(K )| ≥ 3 odd :∑
e∈E(K)
|L(e)|−1 ≤ 12 (|V(K )| − 1) · (1 − δ)
then there exists a proper colouring of the edges of G
so that each edge gets colours from its own listDistance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Kahn’s approach for our case
we have a multigraph F e :
F e
so that each edge e = x1x2 has a list L(e) of at least
(32 + ε) ∆ − (dG(x1) − dF(x1)) − (dG(x2) − dF(x2)) − C2
colours
and F e satisfies “some edge density condition”
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Extending Kahn’s approach
these conditions guarantee that Kahn’s conditions are
satisfied for F e
=⇒ we can edge-colour F e
�
F F e
=⇒ we can colour the Y -vertices in F
choosing from the left-over colours for each
also : we can deal with the “almost” list-edge colouringDistance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Some open problems
prove the next step :
G planar =⇒ χ(G2) ≤ 32 ∆ + O(1)
for G planar, ∆ ≤ 3 we know :
χ(G2) ≤ 7 ( Thomassen, 2007 )
ch(G2) ≤ 8 ( Cranston & Kim, 2006 )
what is the right upper bound for ch(G2) in this case ?
Wegner’s Conjecture for 4 ≤ ∆ ≤ 7 :
G planar =⇒ χ(G2) ≤ ∆ + 5 ?
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Distance-two colouring and edge-colouring
is there some deep relation between edge-colouring
multigraphs and distance-two colouring graphs ?
our proof uses edge-colouring to answer questions on
distance-two colouring
Theorem ( Shannon, 1949 )
for any multigraph G =⇒ χ′(G) ≤ 32 ∆
Conjecture ( Wegner, 1977 )
for a planar graph G , ∆ ≥ 8 =⇒ χ(G2) ≤ 32 ∆ + 1
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008
Edge-colouring and distance-two colouring
Conjecture 1 ( Vizing, 1975 )
for any multigraph G =⇒ ch′(G) = χ′(G)
Conjecture 2 ( Kostochka & Woodall, 2001 )
for any graph G =⇒ ch(G2) = χ(G2)
Conjecture 1 is known for bipartite graphs ( Galvin, 1995 )
proof doesn’t require knowledge of χ′(G) and ch′(G)
Conjecture 2 is only known for some special graph classes,
usually since we know both χ(G2) and ch(G2)
Distance-Two Colouring of Graphs – GT2008, Sandbjerg, 19 August 2008